Shimura and Teichmüller curves

We classify curves in the moduli space of curves Mg that are both Shimuraand Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely onesuch curve, for $g=2$ and g ≥ 6 there are no such curves. We start with a Hodge-theoretic description of Shimura curves and ofTeichmüller curves that reveals similarities and differences of the twoclasses of curves. The proof of the classification relies on the geometry ofsquare-tiled coverings and on estimating the numerical invariants of theseparticular fibered surfaces. Finally, we translate our main result into a classification of Teichmüllercurves with totally degenerate Lyapunov spectrum.